Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ if $\aleph_{\alpha}$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that $\{w < v : \{w,v\} \in E_{G}\}$ is a clique for each $v\in V_{G}$.